Unit 1

Introduction

  • Human life is competitive
    • Job market, romance, toys, etc.
    • Leads to social conflict
  • Darwin provided a framework for copmetition
    • Individuals inherit different traits, and there are limited resources
    • Some traits are better for capturing resources - leads to evolution through natural selection
  • Human life is also cooperative
    • People work together, shop together, line up together, drive together, etc.
    • Humans are more cooperative than chimpanzees; fights over food, strangers are killed
      • Chimpanzees are the closest genetic relatives to humans; common ancestor was 6-7 million years ago
  • Cooperation is puzzling
    • Helping others can give others advantages and reduce your own chance of success
    • Despite there being many opportunities for people to be selfish, they don’t take them
    • Seems difficult to explain
  • Morality refers to the standards used to influence and judge decisions
    • Acts as a duty to prevent against being selfish
    • Constrains individual decision making
    • Can have cooperation with and without morality
    • Morality includes giving to others at a direct cost to yourself
      • On the other hand, cooperation is working together with others and providing resources to gain the return from everyone’s work
  • Humans are the only (observed) species with morality
    • Other species have forms of cooperation; ant, chimpanzees
    • Human parents teach children morals, allowed for by speech and advanced cognitive abilities
    • Utilitarian consequentialism: The right action is the one that produces the most good for the most people
    • Deontological ethcis: Actions should conform with behaviors that serve as universal laws
    • Darwin showed that all humans blush - suggests that shame is universal and morality came about long ago in humankind’s evolutionary past
  • Morality seems to be contradictory to natural selection
    • Selfishness increases your chance of survival, so selfish people should have an advantage
    • The puzzle of morality is that humans are moral despite natural selection seemingly going against morality
  • Millions of years ago, we used to live in a dominance hierarchy
    • Dominance hierarchy: Individuals are ranked highest to lowest; higher-ranked people use their strength to control access to food and mating opportunities, and lower-ranked people must wait
    • Hinders cooperation since long-term research and production is disincentivized due to bullies being able to take away from lower-ranked individual
  • Homo Sapiens lived in a reversed dominance hierarchy
    • Bullies were policed and suppressed by group members (without formal police and courts)
    • Homo Sapiens were more cooperative than modern apes; food sharing, hunting game, supporting children and the injured
    • New cooperation is possible due to protection; can hunt more nutritious game and males are more likely to invest in their children due to more monogamy
    • Other Homo species had the reversed dominance hierarchy
  • Hominin species evolved to become bipedal, less conflict (for mates), stone tools, weapons

Course Goals

  • Study of proximate causes and ultimate causes
    • Proximate causes are immediately responsible for some behavior or event
      • Individual level, e.g. an individuals preferences and beliefs
    • Ultimate causes are deeper forces that explain proximate causes
      • Population level, result of evolutionary processes
        1. Is it instrumentally rational to be cooperative?
    • Instrumental: Acting intentionally to achieve certain outcomes
    • Only outcomes matter
      1. Why are humans so cooperative compared to our closest cousin species?
    • Two key proximate forces: Cooperative dispositions and morality and advanced cognitive abilities
    • Humans are able to predict others’ actions, reflect on their decisions, and discuss morals
    • Leads to suppression of bullies in hunter-gatherer groups and law in modern times
      1. How did human cooperation and morality evolve?
    • Evolutionary pressures led to changes along the hominin line
    • Advances in cooperation and morality allowed for selection of cognitive ability and morality
  • Game theory provides tools to study cooperation and morality
    • Classical game theory explains proximate causes, evolutionary game theory explains ultimate causes
    • Will study rational deliberation, reciprocity in repreated interaction, assortative matching, in-group favoritism, group selection
  • Proximate question: Why are humans so cooperative?
  • Ultimate question: Why did humans become so cooperative?

10 Precepts

  1. Humans manifest unique and puzzling forms of cooperation and morality
  2. Cooperation and morality can be studied using game theory
  3. Selfish motivations generate social dilemmas in which individual actions undermine social welfare
  4. Homo Economicus can achieve self-enforcing cooperation using rewards and threats with a sufficient chance of future interaction
    • Homo Economicus is the representation of a selfish, rational human
  5. Self-enforcing cooperation in repeated interaction among Homo Economicus extends to settings with indirect reciprocity, exclusion, and imperfect monitoring
  6. Human behavior is better represented by Homo Normist than Homo Economicus because it balances conditional rule following and selfishness
    • Homo Normist is a conditional norm follower; they will follow rules if others will
  7. Norm following cannot be explained by consequentialism
    • Norm followers must care about rules, not just outcomes
  8. Evolution can select for norm-following preferences and conditionally-cooperative norms
  9. Positive assortative matching generates a strong evolutionary advantage for cooperators
  10. Evolutionary selection favors norms of in-group favoritism and highly-cooperative groups in group competition

Games and Notation

  • Basic game: Prisoner’s Dilemma
    • Highlights Puzzle of Cooperation due to players being disincentivized to cooperate; Nash equilibrium is Pareto inefficient
    • Games are strategic situations: multi-person, interactive-decision setting where one individual’s actions affect another’s well-being
  • Normal (Strategic) Form: Describes a game using three things
    • A set of individuals (players)
    • A set of actions (strategies) for each player
    • Player’s preference over outcomes resulting from the choices
    • A game matrix captures all three parts
      • Limited in scope; impossible to represent games with infinite choices
    • The Nash equilibrium occurs when all actors have a best response in the same cell of the game matrix
    • Normal-form games are one-shot games where players choose their strategy and follow through with it
  • Notation for a normal form game
    • Set of players: $I = {1, 2, \ldots, n}$
    • Player $i$ chooses a strategy $s_i$ from a set of strategies $S_i$
    • Each player $i$ has a utility $u_i(s)\in \mathbb{R}$ for each ${s = (s_1, s_2, \ldots, s_n)}$ from ${S=S_1 \times \ldots \times S_n}$
    • Shorthand for $s$: $s = (s_i, s_{-i})$, where $-i$ represents all players other than $i$
  • Game theory assumes that invidiuals act rationally
    • Well-defined goals, well-defined utility function
    • Rational agents is known as a methodological assumption used to study human behavior
    • There are rational preferences (consistent choices) and rational beliefs (evidence-based beliefs)
    • Behavior can be instrumental or non-instrumental, selfish or unselfish
      • Homo Economicus is the main model for agents; they are instrumental and selfish, or intentional, consequential, and selfish
      • Some actors might not be Homo Economicus, but they will always be assumed to be rational
  • Pure strategies are strategies that are selected at the beginning and followed without deviation; nonrandom
  • Mixed strategies are strategies in which players “mix” over pure strategies, choosing pure strategies with random probabilities
    • Denoted as $\sigma_i$; if $S_i = {s_{i1}, \ldots, s_{im}}$, then $\sigma_i = (p_{i1},\ldots,p_{im})$, where $p_{ij}$ is the probability that player $i$ chooses strategy $j$
    • $\sum_j p_{ij} = 1$
    • Pure strategies can be thought of as a subset of a mixed strategy where all probabilities (except for one) are set to 0
  • Von Neumann Morgenstern preferences state that rational players maximize their expected utility
    • The best mixed strategy maximizes the expected utility; assumes that mixed strategies are independent from each other
    • Expected utilities require cardinal utilities
  • Solutions in games are special strategies
    • Predicts what will be played
    • Empirically valid; played in reality
    • Mathematically salient; has special properties
      • Defined by a solution concept that has mathematical criteria
      • There are different solution concepts used for different games
  • The best response (BR) for an individual is defined as $s_i^$ such that ${u_i(s^i, s{-1})\geq u_i(s’i, s{-1}), \forall s_i’\in S_i \backslash s_i^*}$
    • Solutions in classical game theory assumes that individuals play best responses
    • BRs can vary based on other players’ strategies
  • A strategy strictly dominates another strategy if the utility for the dominating strategy is always better than the utility for the dominated strategy
    • A strictly dominant strategy is defined as ${u_i(s^_i, s_{-1}) > u_i(s’_i, s_{-1}), \forall s_i’\in S_i \backslash s_i^}$
    • A strictly dominated strategy is defined as ${\exist s’i\in S_i \text{ s.t. } u_i(s^*_i, s{-1}) < u_i(s’i, s{-1}),}$
    • A dominant-strategy solution is when all players have a dominant strategy
  • Iterative Elimnation of Dominated Strategies (IEDS) is a strategy to find solutions to games
    1. Find dominated strategies
    2. Remove them from the pool of options, as they should never be chosen
    3. Repeat steps 1 and 2 until there are no more dominated strategies
  • A game is dominance solvable if IEDS yields a single strategy profile
  • IEDS requires Common Knowledge of the Game (CKG): all players know the structure of the game, and all players know that all other players know the structure of the game
    • CKG is always assumed
  • Common Knowledge of Rationality: Each player knows that other players choose best responses, and all players know that all others know that others choose best responses
    • Without CKR, players are unsure if dominated strategies will be elimated
  • CKR and CKG are higher-order beliefs
  • A strategy profile $s^$ is a (pure) Nash Equilibrium (NE) if ${\forall i\in I, s’_i\in S_i\backslash s^i, u_i(s^*_i, s^*{-i})\geq u_i(s’i, s^*{-i})}$
    • A NE is a strategy profile with mutual BRs; everyone is playing a best response
    • Games can have 0 to many NE, and some games can have non-pure NE with mixed strategies
    • A dominant-strategy solution must be a NE, but not every NE is a dominant-strategy solution
    • NE properties
      • Each player is a rational actor choosing a BR
      • NE can be reached using IEDS
      • Players in a NE do not have deep regret
      • Different processes can lead to NE
      • Nash proved that all solutions have an NE, either mixed or pure
      • NE has predictive power
    • NE shortcomings
      • Might only have mixed NE
      • Must choose one NE; equilibrium sleection
      • Instrumental players must have correct beliefs about others’ strategies
        • This means that experimentally, people do not reach NE
  • A mixed-strategy profile $\sigma^$ is a Mixed Nash Equilibrium iff ${\forall i\in I, \sigma’_i \in \Delta(S_i)\backslash \sigma_i^, u_i(\sigma^_i, \sigma^{-i}) \geq u_i(\sigma’_i, \sigma^*{-i})}$
    • Players should only mix over two strategies if they give the same expected utility; use this fact to find the mixed NE
    • A mixed NE can dominate a pure NE, allowing for IEDS

Social Dillemas

  • Social dilemmas are social interactions (games) in which individual incentives result in an inferior social outcome for the players (Pareto inefficient)
  • Games like the Pure Coordination Game have no conflicts of interest for either player; both gain utility simultaneously
  • Games like the Matching Pennies Games have a pure conflict of interest; for one player to gain utility, the other has to lose utility
  • Games like the Prisoner’s Dilemma has both coordination and conflict; represents a social dilemma
    • Can also be thought as a mixed-motive game; players have an incentive to ciirdubate abd move towards an efficient optimum, but once there, they also have an incentive to shirk

Public Good Game

  • Set of players $I = {1, 2, \ldots, n}$
  • Each player chooses $s_i \in S_i \text{ s.t. } S_i = [0, \bar{s}]$
  • Each player has a utility function $u_i = m\sum_{j\in I}s_j + (\bar{s}-s_i)$
    • Can be thought of as the marginal returns to the total contributions plus the player’s remaining budget
    • Can be rewritten as $u_i = m\sum_{j\neq I}s_j + \bar{s} - (1-m) s_i$
  • Each player’s best response is to set $s^*_i = 0$, as an increase to $s_i$ decreases $u_i$
    • Total social utility in the NE is $U^* = n\bar{s}$
  • The social optimum comes from each player choosing $s_i=s$, making the total social utility $U = \sum (m\sum s + (\bar{s} - s)) = n(mn-1)s + n\bar{s}$
    • $U$ is increasing in $s$ if $n(mn-1)>0 \rightarrow m>\frac{1}{n}$
    • If $m>\frac{1}{n}$, then the social optimal strategy is to contribute $\bar{s}$, making the social optimal utility $mn^2\bar{s}$, much larger than the NE social utility
  • Main issue is the free-rider problem; the individual marginal return is lower than the individual marginal cost, so no one is incentivized to contribute

Tragedy of the Commons

  • Has two players and a common property of size $y>0$
  • Each $i\in I$ chooses how much to consume today (denoted by $c_i\in [0, \frac{y}{2}]$) and then split what is leftover tomorrow to maximize their utility
    • $u_i = \ln c_i + \ln(\frac{y-c_i-c_j}{2})$
    • The first order condition implies that the BR function is $c_i^* = \frac{y-c_j}{2}$
    • NE becomes (c1∗,c2∗)=(y3,y3)(c_1^*, c_2^*) = (\frac{y}{3}, \frac{y}{3})
    • Both players consume $\frac{y}{3}$ today and $\frac{y}{6}$ tomorrow
  • Socially optimal strategy is to consume $\hat{c}_i = \frac{y}{4}$ today; both players consume $\frac{y}{4}$ today and $\frac{y}{4}$ tomorrow
    • Better than the NE total social utility
  • Main issue is the Tragedy of the Commons; individuals are incentivized to over-consume today because everyone else will, hurting their utility tomorrow
    • Gets worse as more players are added

Contests

  • Two individuals $i\in {1,2}$ and efforts $e_i\in \mathbb{R}^+$
  • Probability of winning $p_i(e_i, e_j) = \begin{cases}\frac{e_i}{e_i + e_j}, & e_i + e_j > 0\ \frac{1}{2}, & e_i + e_j = 0\end{cases}$
  • Prize value $v>0$
  • Utility $u_i = p_i(e_i, e_j)v - e_i$
  • BR function: $e_i^* = \sqrt{ve_j} - e_j$
    • NE becomes $e_i^* = \frac{v}{4}$
  • The NE utility for $i$ is $\frac{v}{4}$; the contest intensifies as the prize increases
  • The socially optimal solution is to choose $\hat{e}_i = 0$, as the utility for both players will be $\frac{v}{2}$
    • Another way to see this is by looking at the prize vs. effort exerted; total prize is $v$, but the contestants exert a total of $\frac{v}{2}$ effort, so they effectively “pay” for half of the prize

Chicken Game

  • Two players decide whether or not to fight one another, can either be a chicken or a fighter
  • If both players fight, they both suffer large damage; if neither fights, then they both survive with no loss
  • The two pure NE are for one player to fight and another to be a chicken
  • Can be thought of as a type of contest where you can either give full effort or none
    • Leads to the chicken game also being a coordination game whereas the contest is not

Stag Hunt Game

  • Also known as the Assurance Game
  • Basic context: multiple people are hunting a stag, but everyone needs to be focused to kill the stag; if one player goes off to kill a hare, then no one gets the stag (stag gives more utility than hare)
    • Coordinating to kill the stag is Pareto optimal, but you can only choose to kill the stag if you have assurance, or trust, that the others will also kill the stag
  • Players will only kill the stag if they payoff for the stag is vastly greater
  • Assuming player 2 is playing a mixed strategy, player 1 will choose to kill the stag if and only if $q > \frac{1}{x}$, where $q$ is the chance player 2 will choose $S$ and $x$ is the payoff of $S$
  • The risk-dominant strategy is to mix and choose $S$ with probability $\frac{1}{x}$
  • Risk-averse players will play a maxmin strategy (maximizing the minimum payoff), so they will always choose $H$
  • Assurance games are social dilemmas due to the lack of assurance